TR-2007011: Numerical Computation of Determinants with Additive Preconditioning

Various geometric and algebraic computations (e.g., of the convex hulls, Voronoi diagrams, and scalar, univariate and multivariate resultants) boil down to computing the sign or the value of the determinant of a matrix. For these tasks numerical factorizations of the input matrix is the most attractive approach under the present day computer environment provided the rounding errors are controlled and do not corrupt the output. This is the case where the input matrix is well conditioned. Ill conditioned input matrices, however, frequently arise in geometric and algebraic applications, and this gives upper hand to symbolic algorithms. To overcome the problem, we apply our novel techniques of additive preconditioning and combine them with some nontrivial symbolic-numerical techniques. We analyze our approach analytically and demonstrate its power experimentally.